Time-domain Numerical Study On A Three Dimensional Body Undergoing Large-Amplitude Motion In Waves

On basis of body nonlinear theorem, a free-surface Green function is adopted to solve three-dimensional large-amplitude-body-motion problems in the time domain. The research emphasis is focused on the algorithm of convolution involving time-domain Green function and the analysis of nonlinear body surface effects on the computational results. In addition, the numerical model is expanded to finite depths, constant forward speeds and mooring systems.Traditional formulations of the wave-body interactions assume small amplitude waves and body motions, and the boundary conditions are approximately satisfied on the mean position of the body and free surface by Taylor series expansion. This study focuses on hydrodynamics problems where a large amplitude motion response is induced by small amplitude incident waves. Therefore the exact body boundary condition is satisfied on the instantaneous wetted surface of the moving body, and the free surface condition is linearized. Based on potential flow assumption, the velocity potential satisfies3D Laplace equation. Using Green’s second theory and transportation theorem, boundary integral equation is established by introducing a time-domain free-surface Green function. This equation is applicable to linear and large amplitude body motions.The main difficulty of a time-domain Green function method is that the integral equation involves convolution of time-domain Green function and velocity potential, in which computation and memory requirements grow quickly with time. In this paper, Fourier relationship between time-domain and frequency-domain Green functions (or their derivatives) is obtained. Based on convolution theory, the convolution involving time-domain Green function is replaced by the product involving corresponding frequency-domain Green function. It is so called ’Time-frequency transform method’, and therefore a new boundary equation is established. This equation is applicable to linear or horizontal large amplitude body motion (including large amplitude motions of submerged bodies) problems where the instantaneous wetted body surface is invariable, and provides an efficient way to perform a long-time numerical simulation because of the constant associated computational cost over time.Since free surface Green function is an infinite integral with oscillating integrand function, it is of importance to develop an accurate and efficient algorithm in hydrodynamic computations. On basis of series and asymptotic expansion, time domain and frequency domain Green function are computed for infinite and finite depth, and they are further expanded in Chebyshev polynomials to speed up the computation.In order to improve efficiency and accuracy, a higher-order boundary method is adopted to discretize boundary integral equation. A triangle polar coordinate transformation technique and an auxiliary Green function method are used to eliminate the resulting singularities and ’solid angle’. For an arbitrary large amplitude motion problem of a floating body, the computational domain is variable with time and a stretching dynamic grid technique is introduced to re-mesh the instantaneous wetted body surface. The hydrodynamic and motion equation are solved simultaneously by a dynamic boundary condition. Rigid motion equation is integrated by standard four-order Runge-Kutta method.Numerical computations are first carried out for linear problems of a floating hemisphere and a Wigley hull. The computed forces and motion responses agree well with the frequency-domain solutions and experimental measurements. Second, for large amplitude forced motion problems, numerical results are presented for a submerged sphere and a floating truncated cylinder. The nonlinear body surface effects on the results are analyzed by comparison with the analytical and frequency-domain solutions. Last, the present numerical model is applied to the simulation of large amplitude free motion problems of a floating sphere and a mooring JIP Spar. The nonlinear effects of wave characteristics on the wave forces, motion responses and mooring forces are systemly analyzed.